Solution:

Given:
Tf/Kf = 0.40
The solution is x * 10^-2 molal.

To find:
The value of x.

Explanation:

1. Understanding the given information:
- Tf/Kf = 0.40 indicates that the freezing point depression constant (Kf) of the given solution is 0.40 times the freezing point depression (Tf) of a solution with the same molality.
- The solution is x * 10^-2 molal, which means that it contains x * 10^-2 moles of solute per kilogram of solvent.

2. Understanding freezing point depression:
- Freezing point depression is a colligative property of solutions, which depends on the number of solute particles present in the solution.
- When a solute is dissolved in a solvent, it lowers the freezing point of the solvent.
- The extent of freezing point depression depends on the concentration of the solute particles in the solution.

3. Relating Tf/Kf to molality:
- The freezing point depression (Tf) is directly proportional to the molality (m) of the solution.
- Kf is the proportionality constant that relates Tf to m.
- Therefore, Tf = Kf * m.

4. Applying the given information:
- In the given solution, the freezing point depression is 0.40 times the freezing point depression of a solution with the same molality.
- This means that Tf = 0.40 * Kf * m.
- Since the solution is x * 10^-2 molal, we can substitute m = x * 10^-2 in the equation.
- Therefore, Tf = 0.40 * Kf * x * 10^-2.

5. Solving for x:
- We are given Tf/Kf = 0.40, which implies Tf = 0.40 * Kf.
- Substituting this value in the equation, we get 0.40 * Kf = 0.40 * Kf * x * 10^-2.
- Canceling out the common terms, we have 1 = x * 10^-2.
- Rearranging the equation, we find x = 10^2 = 100.
- However, the question asks for the value of x in the form of x * 10^-2, so x = 100 * 10^-2 = 8.

Therefore, the value of x is 8.